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	<h1>A logistic growth curve model with random effects; <br>
	A comparison with NLME</h1>
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             <font face="Arial, Helvetica" color="White"><b>ADMB Files</b></font>
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                     Code: <a href="orange.tpl">orange.tpl</a><br>
                     Data: 		<a href="orange.dat">orange.dat</a><br>
                     Initial values: <a href="orange.pin">orange.pin</a><br>
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                     All required files (DOS): <a href="orange.zip">orange.zip</a><br>
					 All required files (linux): <a href="orange.tar.gz">orange.tar.gz</a><br>					 
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                     Expected Results: <a href="orange-expected-results.par">orange.par</a><br>
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                     In a <a href="../admb_tutorial.html">DOS</a> window<br> 
					 Under <a href="../admb_tutorial.html">linux</a><br>
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             <font face="Arial, Helvetica" color="White"><b>Results: Computation times</b></font>
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                     ADMB-RE: 0.58 seconds.<br>
					 <TT>nlme</TT>: 1.6 seconds under S-Plus 6.1.<br>
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<h3><strong>Model description</strong></h3>

A growth curve model was fitted to the "orange tree" data by <A HREF="../citations.html#pinh:bate:2000">Pinheiro &amp; Bates (2000, Ch.8.2)</A> 
as an illustration of the <a HREF="http://www.r-project.org/">R</a> (S-Plus) routine <a HREF="http://nlme.stat.wisc.edu/"><TT>nlme</TT></a>. 
The logistic growth curve is given as <br>
<br>
<div align="center"><em>y</em> = <FONT FACE="Symbol">f</FONT><sub>1</sub>
/(1 + exp[-(<em>t</em>-<FONT FACE="Symbol">f</FONT><sub>2</sub>)/<FONT FACE="Symbol">f</FONT><sub>3</sub>])
] + <FONT FACE="Symbol">e</FONT>,
</div>
<br>
where <em>y</em> is the response and <em>t</em> is the age of the three. 
The regression parameters to be estimated are: <FONT FACE="Symbol">f</FONT><sub>1</sub>, <FONT FACE="Symbol">f</FONT><sub>2</sub> 
and <FONT FACE="Symbol">f</FONT><sub>3</sub>, and <FONT FACE="Symbol">e</FONT> is the residual error term. A random effect <em>u</em> is added to the parameter
<FONT FACE="Symbol">f</FONT><sub>1</sub>. A full description of the model can be found here:
 <a href="orange.pdf">orange.pdf</a>.<br><br>
 
<h3><strong>Comparison with <TT>nlme</TT></strong></h3>
Parameter estimates are shown in the following table.
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT"><FONT FACE="Symbol">f</FONT><sub>1</sub></TD>
<TD ALIGN="LEFT"><FONT FACE="Symbol">f</FONT><sub>2</sub></TD>
<TD ALIGN="LEFT"><FONT FACE="Symbol">f</FONT><sub>3</sub></TD>
<TD ALIGN="LEFT">SD(<FONT FACE="Symbol">e</FONT>)</TD>
<TD ALIGN="LEFT">SD(<em>u</em>)</TD>
</TR>
<TR><TD ALIGN="LEFT">ADMB-RE</TD>
<TD ALIGN="LEFT">192.1</TD>
<TD ALIGN="LEFT">727.9</TD>
<TD ALIGN="LEFT">348.1</TD>
<TD ALIGN="LEFT">7.843</TD>
<TD ALIGN="LEFT">31.65</TD>
</TR>
<TR><TD ALIGN="LEFT">Std. dev.</TD>
<TD ALIGN="LEFT">15.658</TD>
<TD ALIGN="LEFT">35.249</TD>
<TD ALIGN="LEFT">27.08</TD>
<TD ALIGN="LEFT">1.013</TD>
<TD ALIGN="LEFT">10.26</TD>
</TR>
<TR><TD ALIGN="LEFT"><TT>nlme</TT></TD>
<TD ALIGN="LEFT">191.0</TD>
<TD ALIGN="LEFT">722.6</TD>
<TD ALIGN="LEFT">344.2</TD>
<TD ALIGN="LEFT">7.846</TD>
<TD ALIGN="LEFT">31.48</TD>
</TR>
</TABLE>
</DIV>

The difference between the estimates obtained with ADMB-RE and <TT>nlme</TT>
is small. The difference is caused by the fact that the two approaches use
different approximations to the likelihood function. (ADMB-RE uses the
Laplace approximation, and for <TT>nlme</TT> the reader is referred to 
(<A HREF="../citations.html#pinh:bate:2000">Pinheiro &amp; Bates, 2000, Ch. 7</A>).

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